The Quantum Tetrahedron and the 6j symbol in quantum gravity

This week as well as working on calculations and modelling the quantum tetrahedron in Lorentzian 3d quantum gravity I have been reading more about the Wigner{6j} symbol, in particular a great paper: Quantum Tetrahedra by Mauro Carfora .

Tetrahedra and 6j symbols in quantum gravity

The Ponzano–Regge asymptotic formula for the 6j symbol and  Regge Calculus are the basis of all discretized approaches to General Relativity, both at the classical and at the quantum level.

sagemath 13 equ0

In Regge’s approach the edge lengths of a triangulated spacetime are taken as discrete counterparts of the metric, a tensor  which encodes the dynamical degrees of freedom of the gravitational field and appears in the classical Einstein–Hilbert action for General Relativity through its second derivatives combined in the  Riemann scalar curvature.

A Regge spacetime is a piecewise linear  manifold of dimension D dissected into simplices; triangles in D = 2, tetrahedra in D = 3, 4 simplices in D = 4 and so on. Inside each simplex either an Euclidean or a Minkowskian metric can be assigned: manifolds obtained by gluing together D–dimensional simplices acquire a metric of Riemannian or Lorentzian signature 2.The Regge action is given explicitly by ( G = 1)

sagemath 13 equ1

where the sum is over (D − 2)–dimensional simplices (hinges), the Vol(D−2)are their (D − 2)–dimensional volumes expressed in
terms of the edge lengths and the deficit angles. A discretized
spacetime is flat inside each D–simplex, while curvature is concentrated at the hinges.  The limit of the Regge action  when the edge lengths become smaller and smaller gives the usual Einstein–Hilbert action for a spacetime which is smooth everywhere, the curvature being distributed continuously.

Regge equations –the discretized analog of Einstein field equations– can be derived from the classical action by varying it with respect to the dynamical variables, i.e. the set  of edge lengths , according to Hamilton principle of classical field theory – see the post: Background independence in a nutshell: the dynamics of a tetrahedron by Rovelli
Regge Calculus gave rise to an approach to quantization of General Relativity known as Simplicial Quantum Gravity. The quantization procedure most commonly adopted is the Euclidean path–sum approach, which is a discretized version of Feynman’s path integral describing D–dimensional Regge geometries undergoing quantum fluctuations.

The Ponzano–Regge asymptotic formula for the 6j symbol;

sagemath 13 equ0

represents the semiclassical limit of a path–sum over all quantum fluctuations, to be associated with the simplest 3–dimensional ‘spacetime’, an Euclidean tetrahedron T. In fact the argument in the exponential reproducesthe Regge action S3 for T.

In general, we denote by T3(j) (a particular triangulation of a closed 3–dimensional Regge manifoldM3 (of fixed topology) obtained by assigning SU(2) spin variables {j} to the edges of T 3. The assignment must satisfy a number of conditions,  illustrated if we introduce the state functional associated with T3(j), namely

sagemath 13 equ2

where No, N1, N3 are the number of vertices, edges and tetrahedra in T3(j).The Ponzano–Regge state sum is obtained by summing over triangulations corresponding to all assignments of spin variables {j} bounded by the cut–off L.

sagemath 13 equ3

where the cut–off is formally removed by taking the limit in front of the sum.

The state sum ZPR [M3] is a topological invariant of the manifold M3, owing to the fact that its value is actually independent of the particular triangulation, namely does not change under suitable combinatorial transformations. These moves are expressed algebraically in terms of  the
Biedenharn-Elliott identity  –representing the moves (2 tetrahedra) <-> (3 tetrahedra)– and of both the Biedenharn–Elliott identity and the orthogonality conditions  for 6j symbols, which represent the barycentric move together its inverse, namely (1 tetrahedra) <-> (4 tetrahedra).

A well–defined quantum invariant for closed 3–manifolds3 based on representation theory of a quantum deformation of the group SU(2)is given by

sagemath 13 equ4
where the summation is over all {j} labeling highest weight irreducible representations of SU(2)q (q = exp{2i/r}, with {j = 0, 1/2, 1 . . . , r − 1}).

The Wigner 6j symbol and its symmetries – The features of the ‘quantum tetrahedron’

Given three angular momentum operators J1, J2, J3 –associated with three kinematically independent quantum systems– the Wigner–coupled Hilbert space of the composite system is an eigenstate of the total angular momentum

J1 + J2 + J3.= J

and of its projection Jz along the quantization axis. The degeneracy can be completely removed by considering binary coupling schemes such as;

(J1 + J2) + J3 and J1 + (J2 + J3),

and by introducing intermediate angular momentum operators defined by;

(J1 + J2) = J12; J12 + J3 = J
(J2 + J3) = J23; J1 + J23 = J

respectively. In Dirac notation the simultaneous eigenspaces of the two complete sets of commuting operators are spanned by basis vectors

|j1j2j12j3> and |j1j2j3j23>

where j1, j2, j3 denote eigenvalues of the corresponding operators, and j is the eigenvalue of J and m is the total magnetic quantum number with range −j < m < j in integer steps.

The j1, j2, j3 run over {0, 1/2 , 1, 3/ 2 , 2, . . . } (labels of SU(2) irreducible representations), while;

|j1 −j2| < j12 < j1 +j2


|j2 −j3| < j23 <j2 + j3.

The Wigner 6j symbol expresses the transformation between the two schemes;

sagemath 13 equ11
Apart from a phase factor, it follows that the quantum mechanical probability;

sagemath 13 equ12

represents the probability that a system prepared in a state of the coupling scheme;

(J1 + J2) = J12; J12 + J3 = J

will be measured to be in a state of the coupling scheme;

(J2 + J3) = J23; J1 + J23 = J

The 6j symbol may be written as sums of products of four Clebsch–Gordan coefficients or their symmetric counterparts, the Wigner 3j symbols. The relations between 6j and 3j symbols are given by;

sagemath 13 equ13
The 6j symbol is invariant under any permutation of its columns or under interchange the upper and lower arguments in each of any two columns. These algebraic relations involve
3! × 4 = 24 different 6j with the same value. The 6j symbol is naturally endowed with a geometric symmetry, the tetrahedral symmetry. In the three–dimensional picture introduced by Ponzano and Regge the 6j is thought of as a real solid tetrahedron T with edge lengths;

L1 = a+ 1/2, L2 ,2 = b+ 1/2 …L6 = f + 1/2

This implies that the quantities;

q1 = a+b+c, q2 = a+e+f, q3 = b+d+f, q4 = c + d + e

(sums of the edge lengths of each face) are all integer. The conditions addressed are sufficient to guarantee the existence of a non–vanishing 6j symbol, but they are not enough to ensure the existence of a geometric tetrahedron T living in Euclidean 3–space with the given edges. More precisely, T exists only if its square volume V evaluated by means
of the Cayley–Menger determinant, is positive.

Ponzano–Regge asymptotic formula
The Ponzano–Regge asymptotic formula for the 6j symbol reads;

sagemath 13 equ0

where the limit is taken for all entries >> 1 and Lp=  j + 1/2,with {jr} = {a, b, c, d, e, f}. V is the Euclidean volume of the tetrahedron T and theta is the angle between the outer normals to the faces which share the edge r.

From a quantum mechanical viewpoint, the above probability amplitude has the form of a semiclassical (wave) function since the factor sagemath 13 equ14ais slowly varying with respect to the spin variables while the exponential is a rapidly oscillating dynamical phase. This  asymptotic behaviour complies with Wigner’s semiclassical estimate for the probability;

sagemath 13 equ14b
compared with the quantum probability:

sagemath 13 equ12

According to Feynman path sum interpretation of quantum mechanics , the argument of the exponential  must represent a classical action, and  it can be read as

sagemath 13 equ14cfor pairs (p, q) of canonical variables, angular momenta and conjugate angle.


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